88324 control system - solution assignment 7 - 2010
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8/9/2019 88324 Control System - Solution Assignment 7 - 2010
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Assignment 7
P6.4
(a) The closed loop characteristic equation is
3 2
( 40)1 ( ) 1 0
( 10)( 20)
30 200 40 0
K sGH s
s s s
or
s s s Ks K
++ = + =
+ +
+ + + + =
The Routh array is
3
2
1
0
1 200
30 40
200 03
40
K s K s
K s
s K
+
Therefore, for stability we require 200 - K/3 > 0 and 40K >0. So, the range of K for stability is0
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The Routh array is3
2
1
0
1 392
24 1099266 0
10992
s
s s
s
There are two sign changes in the first column indicating two roots to right of s = -2.Combining the results, we determine that there are two roots located between s = -1and s = -2. The roots of the characteristic equation ares1 = -27.6250 and s 2,3 = -1.1875 20.8082j
We see that indeed the two roots s 2,3 = -1.1875 20.8082j lie between -1 and -2
P6.7(a) The closed loop characteristic equation is
3 2101 (100 10 ) 100 0a a s s KK s KK + + + + =
The Routh array is3
2
1
0
1 100 10
101 100
100
a
a
a
KK s KK s
b s KK s
+
Where910
100 0.101 a
b KK = + >
Thus examining the first column, we determine that KK a >0 stabilizes the system
(b) The tracking error is
20
100 100( ) lim (1 ( ))
sa
e s s T s s KK
= =
We require E(s)100/0.01745 = 5729
When KKa = 5729, the roots of the characteristic polynomial ares1= -10.15 and s 2,3 = - 45.43 j233.25
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The Routh array is
4
3
2
1
0
1 20 10
7 24 0
11610
7
10
K s
K s
K K s
s b s K
+
Where116
( )(24 ) 707 .
116( )
7
K K K
b K
+ =
Setting b>0 yields2784 - 398K - K 2>0
which holds when- 404.88 < K < 6.876
Examining the first column, we also find that K0 for stability.Combining all the stability regions, we determine that for stability0 0 or 0 < K
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AP6.3)(a) The steady-state tracking error to a step input is
0lim (1 ( )) ( ) 1 (0) 1 ss se s T s R s T
= = =
We want
1 0.05 <
This yields the bounds for 0.95 < < 1.05.the Routh array is
3
2
1
0
1
1 1
0
1
s
sb s
s
+
where
2 11
b
+ =
+
Therefore, using the condition that b>0, we obtain the stability range for :
>0.618
(c) Choosing = 1 satisfies both the steady-state tracking requirement and stabilityrequirement.